The Hawkes process, a self-exciting point process, has a wide range of applications in modeling earthquakes, social networks and stock markets. The established estimation process requires that researchers have access to the exact time stamps and spatial information. However, available data are often rounded or aggregated. We develop a Bayesian estimation procedure for the parameters of a Hawkes process based on aggregated data. Our approach is developed for temporal, spatio-temporal, and mutually exciting Hawkes processes where data are available over discrete time periods and regions. We show theoretically that the parameters of the Hawkes process are identifiable from aggregated data under general specifications. We demonstrate the method on simulated data under various model specifications in the presence of one or more interacting processes, and under varying coarseness of data aggregation. Finally, we examine the internal and cross-excitation effects of airstrikes and insurgent violence events from February 2007 to June 2008, with some data aggregated by day.

翻译：霍克斯过程作为一种自激励点过程，在地震建模、社交网络和股票市场等领域具有广泛应用。传统的估计方法要求研究者能够获取精确的时间戳和空间信息，但实际可用数据往往经过舍入或聚合处理。本文针对聚合数据开发了一种霍克斯过程参数的贝叶斯估计方法。我们的方法适用于时间、时空及互激励霍克斯过程，其中数据以离散时间段和区域的形式呈现。理论上我们证明了在一般设定下，霍克斯过程的参数可以从聚合数据中识别。通过模拟数据，我们在多种模型设定下验证了该方法在单过程或多交互过程场景中的有效性，并考察了不同聚合粒度的影响。最后，我们以2007年2月至2008年6月期间的空袭事件与叛乱暴力事件为研究对象，分析了其内部激励与交叉激励效应，其中部分数据已按日进行聚合。